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In this paper, we consider a non-standard renewal risk model, in which each main claim may produce a random number of delayed claims. Such a model takes full account of lasting impacts of severe claims and hence may avoid the underestimation of an insurer's operation risk within a relatively long period. We obtain asymptotic expansions for the finite-time ruin probability, given that the claim size distributions belong to the general subexponential class. When the claim size distributions are restricted to the smaller class of extended regular variation, we push our study forward to the infinite-time ruin probability.
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